3x=5/6x+13

Solution for 3x=5/6x+13 equation:

3x=5/6x+13

We move all terms to the left:

3x-(5/6x+13)=0


Domain of the equation: 6x+13)!=0

x∈R


We get rid of parentheses

3x-5/6x-13=0

We multiply all the terms by the denominator


3x*6x-13*6x-5=0

Wy multiply elements

18x^2-78x-5=0

a = 18; b = -78; c = -5;
Δ = b2-4ac
Δ = -782-4·18·(-5)
Δ = 6444
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{6444}=\sqrt{36*179}=\sqrt{36}*\sqrt{179}=6\sqrt{179}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-78)-6\sqrt{179}}{2*18}=\frac{78-6\sqrt{179}}{36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-78)+6\sqrt{179}}{2*18}=\frac{78+6\sqrt{179}}{36}
$